Finite Group Databasebeta

← #10819935

Subgroups — #10819935

Order 768.

878 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5092C211yesmaximal, characteristic
2#82549C212yesmaximal
3#78948C212yesmaximal
4C3×#1886C211yesmaximal, characteristic
5#71784C211yesmaximal, characteristic
6#70336C212yesmaximal
7#70108C212yesmaximal
8#68765C212yesmaximal
9#68741C212yesmaximal
10#23200—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C3×#519C2211yescharacteristic
13#5812C2211yescharacteristic
14C22×#216C2212yes—
15C22×#216C2212yes—
16#5516C2212yes—
17C2×#869C2211yescharacteristic
18C3×#419C2212yes—
19C2×C3×#101C2211yescharacteristic
20C2×#836C2211yescharacteristic
21#5225C2212yes—
22C2×#824C2212yes—
23C2×#822C2212yes—
24#5122C2212yes—
25#5120C2212yes—
26#5092C418yescomplemented
27#5030—24no—
28C4×#216—24no—
29#5004C2212yes—
30#4998C2212yes—
31#4997C2212yes—
32C2×C4×C3⋊C8C2211yescharacteristic
33#4790—24no—
34#4789—24no—
35C3×#339C2212yes—
36#4523C2212yes—
37C2×#494—31no—
38#2950—32no—
39#2950—32no—
40#1979—32no—
41#1979—32no—
42#1886—32no—
43#1886—31no—
44#1886D311yescharacteristic
45#1772—32no—
46C22×C3×Q8C2311yescharacteristic
47C22×C3×Q8C2311yescharacteristic
48C22×Dic6C2312yes—
49C2×C3×M16C2312yes—
50C3×C4⋊Q8—41no—
51C3×C4⋊Q8C2×C414yes—
52C3×C4×Q8—24no—
53C3×#117—41no—
54C3×#117C2312yes—
55C2×C3×C4⋊C4—41no—
56C2×C3×C42C2311yescharacteristic
57Q8×Dic3—24no—
58C2×#216—22no—
59C2×#216—22no—
60C2×#216—22no—
61C2×#216—22no—
62C2×#216—24no—
63C2×#216—24no—
64C2×#216C2×C418yes—
65C2×#216C2×C418yes—
66#926—41no—
67#926C2312yes—
68C2×C4⋊Dic3—41no—
69C2×#208C2312yes—
70C22×C3⋊C8C2312yes—
71#887—24no—
72#877—42no—
73#869—41no—
74#869C2×C414yes—
75C4×Dic6—24no—
76C3×#103—24no—
77C3×#101—41no—
78#836—41no—
79#835—24no—
80#824—42no—
81#822—24no—
82#816—24no—
83#815—22no—
84C8×Dic3—22no—
85#810—24no—
86#808D414yes—
87#807D414yes—
88#803—22no—
89C4×C3⋊C8—22no—
90C4×C3⋊C8C2×C414yes—
91C22×Q16—32no—
92C22×Q16—32no—
93C2×C4⋊Q8—31no—
94C2×C4⋊Q8C2×D311yescharacteristic
95#519—31no—
96#519—32no—
97#519C2×D311yescharacteristic
98#494—38no—
99#441—64no—
100#441—64no—
101C4×Q16—64no—
102C4×Q16—64no—
103C2×#108—32no—
104#427—32no—
105#419—32no—
106#419—32no—
107#419—32no—
108#419C2×D312yes—
109C2×#101—31no—
110C2×#101—32no—
111C2×#101C2×D311yescharacteristic
112#405—32no—
113#405—32no—
114C2×C4×C8—31no—
115#339—32no—
116#339C2×D312yes—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8C22×C412yes—
124C2×C3×Q8C22×C412yes—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4C2411yescharacteristic
129C22×Dic3—22no—
130C2×Dic6—22no—
131C2×Dic6—22no—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6—22no—
135C2×Dic6C22×C414yes—
136C3×M16—22no—
137C3×M16—22no—
138C2×C3×C8—22no—
139C3×C4⋊C4—44no—
140C3×C4⋊C4—42no—
141C3×C4⋊C4C2×D414yes—
142C3×C22⋊C4—22no—
143C3×C42—22no—
144C3×C42—21no—
145C3×C42—42no—
146C3×C42C22×C412yes—
147#217—22no—
148#216—28no—
149#216—28no—
150#216—28no—
151#216—28no—
152C4⋊Dic3—44no—
153C4⋊Dic3—42no—
154C4⋊Dic3C2×D414yes—
155Dic3⋊C4—44no—
156C4×Dic3—22no—
157C4×Dic3—42no—
158C4×Dic3—44no—
159#208—22no—
160#208—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8C22×C414yes—
163C2×C3⋊C8C2×D412yes—
164C2×C3⋊C8C2×D412yes—
165C2×C3⋊C8#4114yes—
166C22×Q8—32no—
167C22×Q8C22×D311yescharacteristic
168C22×Q8C22×D311yescharacteristic
169C2×Q16—38no—
170C2×Q16—38no—
171C2×Q16—64no—
172C2×Q16—64no—
173C2×Q16—62no—
174C2×Q16—62no—
175C2×Q16—62no—
176C2×Q16—62no—
177C2×M16—32no—
178C2×M16C22×D312yes—
179C22×C8—32no—
180C4⋊Q8—34no—
181C4⋊Q8—41no—
182C4⋊Q8—121no—
183C4⋊Q8C4×D314yes—
184C4×Q8—24no—
185C4×Q8—64no—
186C4×Q8—64no—
187C4×Q8—64no—
188#117—32no—
189#117—41no—
190#117—122no—
191#117—121no—
192#117C22×D312yes—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—121no—
195C2×C42C22×D311yescharacteristic
196#108—64no—
197#107—34no—
198#106—34no—
199#103—24no—
200#103—64no—
201#103—64no—
202#103—64no—
203#101—41no—
204#101—122no—
205#101—121no—
206#97—62no—
207#97—62no—
208C4×C8—34no—
209C4×C8—62no—
210C4×C8—62no—
211C23×C3H(5,𝔽2)11yescharacteristic
212C3×Q8—24no—
213C3×Q8—24no—
214C3×Q8—42no—
215C3×Q8—42no—
216C2×C3×C4—24no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4C23×C411yescharacteristic
226C2×C3×C4C22×D412yes—
227C2×C3×C4C2×#4112yes—
228C2×C3×C4H(5,𝔽2)11yescharacteristic
229C2×Dic3—24no—
230C2×Dic3—24no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233Dic6—24no—
234Dic6—24no—
235Dic6—42no—
236Dic6—42no—
237C3×C8—42no—
238C3⋊C8—42no—
239C3⋊C8C4×D414yes—
240C3⋊C8C4×D414yes—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—34no—
248C2×Q8—62no—
249C2×Q8—62no—
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8C2×C4×D312yes—
254C2×Q8C2×C4×D312yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—62no—
259C22×C4C23×D311yescharacteristic
260Q16—68no—
261Q16—68no—
262Q16—68no—
263Q16—68no—
264M16—22no—
265M16—22no—
266M16—62no—
267M16—62no—
268C2×C8—22no—
269C2×C8—34no—
270C2×C8—32no—
271C2×C8—32no—
272C2×C8—34no—
273C2×C8—62no—
274C4⋊C4—34no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—124no—
278C4⋊C4—124no—
279C4⋊C4—122no—
280C4⋊C4D3×D414yes—
281C22⋊C4—22no—
282C22⋊C4—62no—
283C42—22no—
284C42—21no—
285C42—42no—
286C42—62no—
287C42—124no—
288C42—122no—
289C42C2×C4×D312yes—
290C22×C3—22no—
291C22×C3#51811yescharacteristic
292C22×C3D4211yescharacteristic
293C22×C3#55911yescharacteristic
294C3×C4—24no—
295C3×C4—42no—
296C3×C4—42no—
297C3×C4—44no—
298C3×C4C2×C4×D412yes—
299C3×C4#55012yes—
300Dic3—44no—
301Dic3—44no—
302C23#100411yescharacteristic
303Q8—24no—
304Q8—24no—
305Q8—42no—
306Q8—42no—
307Q8—64no—
308Q8—64no—
309Q8—122no—
310Q8—122no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—64no—
321C2×C4—64no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4C22×C4×D311yescharacteristic
325C2×C4C2×D3×D412yes—
326C2×C4#100411yescharacteristic
327C2×C4D3×#4112yes—
328C8—34no—
329C8—34no—
330C8—42no—
331C8—122no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3#236611yescharacteristic
335C2×C3#329312yes—
336C22—22no—
337C22#548611yescharacteristic
338C22#559011yescharacteristic
339C22#566411yescharacteristic
340C4—24no—
341C4—44no—
342C4—42no—
343C4—42no—
344C4—124no—
345C4—124no—
346C4C4×D3×D412yes—
347C4#559912yes—
348C3#2320011yescharacteristic, complemented, Sylow
349C2—21no—
350C2—21no—
351C2#7467911yescharacteristic
352C2#8030312yes—
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1886C2
socleC22×C3D42
prime radicalC23×C3H(5,𝔽2)
cosocleC22×C3×C4C24
hypercenter#1886D3
hypocenterC3#23200
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1886

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
26. #5092C4semidirect
348. C3#23200semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#50922
#825492
#789482
C3×#18862
#717842
#703362
#701082
#687652
#687412
#232003